Can someone help me with this question?
A can in the shape of a right circular cylinder is required to have a volume of 1,000 cubic centimeters. The top and bottom are made up of a material that costs 10¢ per square centimeter, while the sides are made of material that costs 5¢ per square centimeter. Find a function that describes the total cost of the material as a function of the radius r of the cylinder.
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To find a function describing the total cost of the material as a function of the radius (r) of the cylinder, we need to determine the dimensions of the cylinder and then calculate the cost of each component.
First, we know that the volume of the cylinder is given by the formula V = πr^2h, where r is the radius and h is the height of the cylinder.
Since we are given that the volume needs to be 1,000 cubic centimeters, we can express this as an equation:
1000 = πr^2h
To make calculations easier, we can solve this equation for h:
h = 1000 / (πr^2)
Now we can determine the cost of each component:
The cost of the top and bottom surfaces is 10¢ per square centimeter. Each of these surfaces is a circle with radius r, so the total cost for both circles is:
Cost_top_and_bottom = 2 * (πr^2) * 10¢
The cost of the sides is 5¢ per square centimeter. The sides consist of a rectangle with height h and length equal to the circumference of the circle at the base of the cylinder:
Circumference = 2πr
The total cost for the sides is:
Cost_sides = (2πr) * h * 5¢
Finally, the total cost of the material is the sum of the costs of the top and bottom surfaces and the sides:
Total_cost(r) = Cost_top_and_bottom + Cost_sides
Plugging in the expressions for each cost, we get:
Total_cost(r) = 2 * (πr^2) * 10¢ + (2πr) * h * 5¢
Now, substitute the value of h we found earlier:
Total_cost(r) = 2 * (πr^2) * 10¢ + (2πr) * (1000 / (πr^2)) * 5¢
Simplifying further, we can cancel out some terms:
Total_cost(r) = 20πr^2 * 10¢ + 10,000π / r * 5¢
Total_cost(r) = 200πr^2¢ + 50,000π / r¢
Thus, the function that describes the total cost of the material as a function of the radius (r) of the cylinder is:
Total_cost(r) = 200πr^2 + 50,000π / r